Characteristic, Counting, and Representation Functions Characterized

Charles Helou

doi:10.1007/978-3-319-68032-3_9

Characteristic, Counting, and Representation Functions Characterized

Charles Helou

Penn State Brandywine

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Abstract

Given a set A of natural numbers, i.e., nonnegative integers, there are three distinctive functions attached to it, each of which completely determines A. These are the characteristic function χ_A(n) which is equal to 1 or 0 according as the natural number n lies or does not lie in A, the counting function A(n) which gives the number of elements a of A satisfying a ≤ n, and the representation function r_A(n) which counts the ordered pairs (a, b) of elements a, b ∈ A such that a + b = n. We establish direct relations between these three functions. In particular, we express each one of them in terms of each other one. We also characterize the representation functions by an intrinsic recursive relation which is a necessary and sufficient condition.

Original language	English (US)
Title of host publication	Combinatorial and Additive Number Theory II - CANT
Editors	Melvyn B. Nathanson
Publisher	Springer New York LLC
Pages	139-155
Number of pages	17
ISBN (Print)	9783319680309
DOIs	https://doi.org/10.1007/978-3-319-68032-3_9
State	Published - Jan 1 2017
Event	13th Workshop on Combinatorial and Additive Number Theory, CANT 2015 - New York, United States Duration: May 19 2015 → May 22 2015

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	220
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Other

Other	13th Workshop on Combinatorial and Additive Number Theory, CANT 2015
Country/Territory	United States
City	New York
Period	5/19/15 → 5/22/15

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1007/978-3-319-68032-3_9

Cite this

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title = "Characteristic, Counting, and Representation Functions Characterized",

abstract = "Given a set A of natural numbers, i.e., nonnegative integers, there are three distinctive functions attached to it, each of which completely determines A. These are the characteristic function χA(n) which is equal to 1 or 0 according as the natural number n lies or does not lie in A, the counting function A(n) which gives the number of elements a of A satisfying a ≤ n, and the representation function rA(n) which counts the ordered pairs (a, b) of elements a, b ∈ A such that a + b = n. We establish direct relations between these three functions. In particular, we express each one of them in terms of each other one. We also characterize the representation functions by an intrinsic recursive relation which is a necessary and sufficient condition.",

author = "Charles Helou",

year = "2017",

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day = "1",

doi = "10.1007/978-3-319-68032-3_9",

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note = "13th Workshop on Combinatorial and Additive Number Theory, CANT 2015 ; Conference date: 19-05-2015 Through 22-05-2015",

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Helou, C 2017, Characteristic, Counting, and Representation Functions Characterized. in MB Nathanson (ed.), Combinatorial and Additive Number Theory II - CANT. Springer Proceedings in Mathematics and Statistics, vol. 220, Springer New York LLC, pp. 139-155, 13th Workshop on Combinatorial and Additive Number Theory, CANT 2015, New York, United States, 5/19/15. https://doi.org/10.1007/978-3-319-68032-3_9

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T1 - Characteristic, Counting, and Representation Functions Characterized

AU - Helou, Charles

PY - 2017/1/1

Y1 - 2017/1/1

N2 - Given a set A of natural numbers, i.e., nonnegative integers, there are three distinctive functions attached to it, each of which completely determines A. These are the characteristic function χA(n) which is equal to 1 or 0 according as the natural number n lies or does not lie in A, the counting function A(n) which gives the number of elements a of A satisfying a ≤ n, and the representation function rA(n) which counts the ordered pairs (a, b) of elements a, b ∈ A such that a + b = n. We establish direct relations between these three functions. In particular, we express each one of them in terms of each other one. We also characterize the representation functions by an intrinsic recursive relation which is a necessary and sufficient condition.

AB - Given a set A of natural numbers, i.e., nonnegative integers, there are three distinctive functions attached to it, each of which completely determines A. These are the characteristic function χA(n) which is equal to 1 or 0 according as the natural number n lies or does not lie in A, the counting function A(n) which gives the number of elements a of A satisfying a ≤ n, and the representation function rA(n) which counts the ordered pairs (a, b) of elements a, b ∈ A such that a + b = n. We establish direct relations between these three functions. In particular, we express each one of them in terms of each other one. We also characterize the representation functions by an intrinsic recursive relation which is a necessary and sufficient condition.

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DO - 10.1007/978-3-319-68032-3_9

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T3 - Springer Proceedings in Mathematics and Statistics

SP - 139

EP - 155

BT - Combinatorial and Additive Number Theory II - CANT

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PB - Springer New York LLC

T2 - 13th Workshop on Combinatorial and Additive Number Theory, CANT 2015

Y2 - 19 May 2015 through 22 May 2015

ER -

Characteristic, Counting, and Representation Functions Characterized

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